The M12-Design

The orbits of
{1, 2, 3, 4, 5, 7},
{1, 3, 4, 7, 10, 11} and
{1, 4, 5, 7, 9, 10}
under A form the blocks of the M12-design.
The stabilizer of first two sets in A is trivial,
the third is of order five, generated by x.
The orbits are thus of length 60, 12 and 60 respectively.

{1, 2, 3, 4, 5, 7}

{1, 3, 4, 7, 10, 11}

{1, 4, 5, 7, 9, 10}

Let i=1,...12 denote the elements of the projective line over GF(11),
where we identify
< (1,i) >
with i , i=1,...,10 and put
< (1,0) > = 11 and
< (0,1) > = 12.

The matrices

6 0
0 2

and

1 9
5 2

induce the given generators x and y on the projective line.
Thus, we embed A into PSL(2,11).
The three orbits indicated above fuse into one orbit of length 132
under PSL(2,11).
The element x generates the stabilizer of
{1, 3, 4, 7, 10, 11}
in PSL(2,11), too.